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As an essential component of the modeling framework presented in [17], interface solid elements (ISE), i.e., degenerated solid finite elements with almost zero thickness proposed by Manzoli et al. [18,19,20,21], have been adopted and successfully applied to the failure analysis of plain and fiber-reinforced concrete structures. As compared with classical zero-thickness interface elements, ISE can be easily implemented based on standard finite element codes by using solid finite elements for the bulk material and for the interfaces. Employing a continuum damage model to approximate the interface degradation, it allows one to describe the interface behavior completely in the continuum framework. Consequently, those specific variational formulations, discrete constitutive relations and integration rules to obtain the internal forces associated with classical interface elements are not required. The artificial initial stiffness that is normally required in zero-thickness interface elements is automatically included in the elastic stiffness of ISE [18,20,21]. It is recognized that the interface solid elements share similar features with zero-thickness interface elements. The most notable advantage of this class of models is the fact that no special procedure for the tracking of evolving cracks is necessary. This contributes to its robustness and allows for 3D fracture simulations characterized by complex fracture patterns (see, e.g., [22]). The crack pattern obtained via discrete representations along prescribed element edges evidently suffers from a certain dependence on the mesh topology. However, the influence on the overall macroscopic material response is tolerable if unstructured meshes with reasonable resolution are used [20,23]. Furthermore, for analyses of heterogeneous materials on the mesoscale level, it was shown that the mesh-dependence of interface elements becomes less of a concern once the mesoscale heterogeneity is modeled [24,25,26]. This drawback can be alleviated, e.g., by continuously modifying the local finite element topology at the crack tip to enforce the alignment between the element edges and the crack propagation direction [27,28,29]. Alternatively, mesh refinement, at the cost of increased computational expense, can be applied to resolve the large elements along the crack path [30,31,32]. The increased computational demand resulting from the duplication of finite element nodes can be controlled by pre-defining the interface elements only in vulnerable regions or applying an adaptive algorithm for the mesh processing during computation [33,34,35,36,37].

Here, ω denotes the relative problem size while using the adaptive algorithm as compared with the case of full fragmentation. With the growth of the macroscopic crack, ISEs are gradually generated and located along the potential crack path. When the applied displacement reaches 0.6 mm, the propagating macroscopic crack almost penetrates the sample. Afterwards, the major crack continues to open, and the structure fails rapidly; the relative problem sizes change marginally, approaching approximately 34% for the number of nodes and 40% for the number of elements, respectively, at the end of simulation when u = 1 mm at increment i = 1000: 2b1af7f3a8